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12-1992 Almost Difference Sets and Reversible Divisible Difference Sets James A. Davis University of Richmond, [email protected]
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Recommended Citation Davis, James A. "Almost Difference Sets and Reversible Divisible Difference Sets." Archiv Der Mathematik 59, no. 6 (December 1992): 595-602. doi: 10.1007/BF01194853.
This Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. Arch. Math., Vol. 59, 595-602 (1992) 0003-889X/92/5906-0595 $ 3.10/0 1992 Birkh/iuser Verlag, Basel
Almost difference sets and reversible divisible difference sets
By JAMES A. DAVIS
1. Introduction. Let G be a group of order mn and N a subgroup of G of order n. If D is a k-subset of G, then D is called a (m, n, k, 21,22) divisible difference set (DDS) provided that the differences dd' - i for d, d' e D, d 4= d' contain every nonidentity element of N exactly 21 times and every element of G - N exactly 22 times. Difference sets are used to generate designs, as described by [4] and [9]. D will be called an Almost Difference set (ADS) if )ol and 22 differ by 1. The reason why these are interesting involves their relationship to symmetric difference sets. A symmetric difference set is a DDS with 21 = 22, so the ADS are "almost" difference sets. Symmetric difference sets are becoming more and more difficult to construct, and this is as close as we can get with a divisible difference set. One helpful way to view DDS is to consider the group ring Z[G]. We will abuse notation by associating to the subset A of G the element A = ~ a of the group ring. a~A Similarly, we will write A (- l) = ~ a- i. The definition of a DDS immediately yields the aeA group ring equation DD (-1) = k + 2i(N - 1) + ,~2(G -- N). This is often a very simple way to check whether a subset of a group meets the properties of being a divisible difference set, and it has the advantage that it works in nonabelian cases. We will use this technique to check the second construction. We now restrict our attention to abelian groups; in this case, characters of the group are simply homomorphisms from the group to the complex numbers (we are following the lead of Turyn [13]). Extending this homomorphism to the entire group ring yields a map from the group ring to a number field. The character sum for the character Z on the element D of the group ring yields 3 possible results: Z(D) = k if )f is the principal (all l) character, Iz(D)l =~,/~-2i if Z is nonprincipal on N, and Iz(D)I= xf k - )u + (21 - '~2) n if)~ is principal on N but nonprincipal on G. One other useful fact about characters is that if Z is a nonprincipal character on a group (or subgroup) H, then the character sum ofz over H will be 0. If we have a subset of the group that satisfies these character sums for every character Z, then that subset will be a DDS (this is because of the orthogonality relations for characters: see [13] for similar arguments). Thus, our strategy will be to come up with a "candidate" subset of the group, then use character theory to check that all the sums are correct. We will use this techniques to check the first construction.
38* 596 J.A. DAWS ARCH. ~a.TH.
One other condition that we wilt note in this paper involves the multiplier group of the difference set. We say that a difference set is reversible ifD (- ~) = D. This is a nice property for a set to have because it has strong implications for all of the other multipliers. For symmetric difference sets, it is conjectured that there is only one non-Menon difference set with multiplier - 1. Reversible DDSs have been the subject of much recent research, with both constructions and nonexistence results appearing in the literature: In particu- lar, Ma [!0, 1t] ha s provided several nonexistence results based on number theoretic considerations on the parameters. Jungnickel [7, 8], and then Arasu, Jungnicket, and Pott [2] have provided constructions of reversible DDSs based on the constructions for re- versible symmetric difference sets found in McFarland's paper [10]. Construction 2 will yield some new reversible DDSs that generalize the constructions found in these papers.
2. Construction 1, The first construction has connections to a familiar set of parameters from symmetric difference sets. Turyn [14] has constructed (4.32~ 2.320 -3~, 3 ~" - 3") in groups of the form H x EA (32~ where H is either group of order 4, and EA (32,) is the elementary abelian group of order 32". In an effort to find other groups that have difference sets with these parameters, this author constructed divisible difference sets; but these divisible difference sets did not have 2~ close to 22 (see [5]). Since then, this author, together with Jedwab, Arasu, and Sehgal have constructed difference sets with the same parameters as Turyn's examples (see [1]). A minor adjustment of that difference, set will yield an ADS. Let G be the group of the form H x Zzo where H is either group of order 4. The generators of the 3 a parts are y and z, and we will write the elements of H as ho, hi, h 2, h 3. The subgroup N is the group (y, z> ~ Z~o. We also want to label the cyclic subgroups 3 '+~ - 1 of order 3 ~ in a careful way. We wilt use D1, ~ = (yz~>, i = 0, 1, ..., 3 -- i - 2 and 3 ~ - I D i,1 = 3 ~ -- 1 3 ~ -- t ...... ! ...... t D= k=o hk i=o~ ziD1,31+k W 3 j~o yiDj'I " This is the "candidate" subset; we claim that this is an ADS with the parameters (4,32~ 2(3 z~ - 3"), 320 - 2.3", 32" - 2.3 ~ + 1). The only difference between this and the difference sets found in [1] is this construction is missing one coset of the D~,j. The proof is broken down into the following sequence of lemmas (these are the essentialIy the same as found in [1], and we do not include the proofs here). I_emma 2.1. D has no repeated elements. This ensures us that we wilt have the correct number of elements in the difference set, and that there is no overlapping of the cosets. Lemma 2.2. If Z is a character of order 3" on G/H, then Z is nonprincipaI on all of the Di, j except one, where it is principal. Vol. 59, 1992 Almost difference sets 597 This lemma declares that any character of order 3" will have a character sum of 0 over all of the Di, j except one. Thus, the character sum over the entire set will be (in modulus) the size of that D~,j, which is 3". The only difficulty in the proof involves cases when the D~,j shows up twice. In that case, the character sum will be 3"7~(z~) (1 + ~3o-1), where ~ is a primitive 3" root of unity. The modulus of that sum is 3 ~ so it matches what is required. The next lemma describes the situation for all of the other characters that are nonprin- cipal on the forbidden subgroup (y, z). Lemma 2.3. If )~ is a character of G/H that is nonprincipal but of order less than 3", then 3 a - 1 3 a - 1 x(ziD1,3i+k) = 2 7~(YJDj,O =0 i=O 1=0 for every k except I, where the character sum is 3" in modulus. Thus, the character sum is correct for all of the characters that are nonprincipal on (y, z). All that remains is to check the character sums for the characters that are principal 32~ __ 3a on (y, z), but nonprincipal on H. Each of the cosets of (y, z) has -- elements, so 32" - 3" 3 2 the character sum is --2 ~=o~ z(hl) = 0 = ,fk - 21 + (2 z - 22) n, which is correct. The next lemma summarizes that result. Lemma 2.4. If Z is a nonprincipal character on G that is principal on (y, z), then )~(D) = O. Putting all this together, we get the following Theorem 2.1. D is a (4,32a, 2(3 2a -- 3~), 32" - 2 3", 3 2a -- 2- 3 a + 1) ADS in G. This DDS generates a divisible design with the same parameters, and that design is semi-regular since k 2 = (2(32" - 3a)) 2 -= 4' 32"(3 a - 1) 2 --= mn2z. The author has not found this set of parameters in the literature, so this seems to be a new set of parameters for a semi-regular design that is also an ADS. 3. Construction 2. Another set of parameters that have been studied extensively for \- q---1 + q ~ q _ 11)' qe\~_l))" The orig- inal construction of these difference sets can be found in McFarland's paper [10]. Dillon [6] modified this construction slightly, and it has been further generalized in [2, 7, 8]. All of these constructions have the same basic strategy: take all of the hyperplanes of an elementary abelian group EA (qa+ 1) (call them H 1, H 2 .... , H~A), and attach a differ- ent coset representative to each hyperplane. We consider one further generalization of this construction; instead of using a difference set (as in [2] and [8]), we will use a DDS with the property that k - 21 = 1. Let A be a (m, n, h, h - 1, 22) DDS in a group H relative to qa+l __ ] a normal subgroup K, where h - If the elements of A are a~, i = 1, 2, ..., h, then h q--1 form the set D = Q) aiHi of HxEA(qa+l). i=1 598 J.A. Davis ARCH. MATH. Theorem 3.1. D is ~ , q , qd q--i ' q~ , qd-122 ) DDS in H x EA(q ~+1) relative to K x EA(qa+I). Proof: h h DD(-1)= ~. aiHi ~. Hja~ 1 i=t j=l h = qd ~ a~Hia[1 + qe-lEA(q~+t ) ~ a~a~t i=1 i*j --,E q -- I (EA(qa+~)- 1)1 ~(qd+l--1 ) + qe-lEA(qa+t) L\ q--1 i (K -- 1) + 22(H - K) _ qd q d + l - 1 d~ q-1 +qd _ ((KxEA(qe+I))-I) + qe-i 22(G _ (K x EA(qe+l))). We had to require that the set A be a DDS with the property k - 21 = 1 in order to satisfy the equation q~- ~ 21 = qe q~ - 1 [] q-i" This construction works even if H and K are nonabelian groups, and it will also work if this is not a direct product group, but we will need to put a Dillon-like condition on the permutation of the hyperplanes. Thus, we need to investigate the DDS that have the property k - 21 = 1. The first case we will consider is called a trivial case. Let H be any group with a normal subgroup K. It is easy to see that the set (H-K) vo{l} is a (m,n,(m-1) n+l, (m-1)n, (m - 1) n - n + 2) DDS in H relative to K. Since this is a DDS with the correct property, we get the following corollary. qa§ _ 1 qe+l _ i q-- 1 Corollary 3.1. Suppose that n is a divisor of 1, and that m = + 1. q--I n If H is any group of order m n with a normal subgroup K of order n, then we will get a (m'nqe+l'qe qq---i ,qd ,qe-l((m-t)n-n+2)) DOS in HxEA(q ~+!) relative to K x EA(q a+ 1). We should make several comments about this construction. First, when n = 2, the parameters reduce to a difference set (since 21 = '~2), and this construction is McFarland's original idea. Second, when n = 1, we get the construction found in [2] and [8]. It is d+l '~ q~+l _ I q~+l semiregular since k 2 _ze//q __- - 1 2 = qa+lq~-i , i _ ran22" It is =q Ik q--t ) q-I q-I not an ADS in general, but it is when d = I (these parameters are found in [2]). Vol. 59, 1992 Almost difference sets 599 Corollary 3.2. If H is a group of order q + 1, then h xEA(q 2) will have a (q + l, q2 q(q + 1),q,q + 1)ADS. Another interesting case occurs when q = 3 a, n = 3, and d = 1: Corollary 3.3. D is a (3 "-1 + 1, 3 2a+1, 3a(3 a -}- 1), 3 ", 3 a -- ]) ADS in G. This is not a semi-regular example, but it is the only example that the author is aware of that has 2 2 = 21 - 1. Finally, we consider an important consequence of this result that does not relate to ADS, we can get reversible DDS. Corollary 3.4. If m and n are both powers of 2, then there is a reversible DDS with the parameters listed above. P r o o f" Let the group be Z~ +v • EA(q d+ 1), where m = 2 u, n = 2 ". Since every element of Z~ +~ is fixed under inversion, and every hyperplane is fixed under inversion, it is clear that D = D (- 1~ D We list several examples of this below: n q d m (m, nq d+ l, qdh, q"~-~-qe-1 l ' qe_122 ) 1 2 5 2 16 (16, 250, 775, 150, 150) 2 1 3 1 4 (4, 9, 12, 3, 4) 3 1 7 1 8[ (8, 49, 56, 7, 8) 4 1 31 I 32 I (32,961,992,31,32) 5 4 3 2 4 (4, t08,117,36,3~ 6 8 7 2 8 (8,2744,2793,392,350) 7 32 31 2 32 (32,953312,954273,30752,29822) 8 8 3 4 16 (16,1944,9801,3240,3078) Notice several things about this list. First, Example ~ 1 is the sporadic example of a (non-Menon) symmetric difference set that has multiplier - 1. Examples ~+ 2-4 come from [2]. Examples #~ 5-7 all have the property that m = n, and q = n - 1. Whenever 2 u - 1 is a prime power, then the corollary will apply to m = n = 2 u, q = 2 z' - 1, d = 2. Thus, we will also have examples of DDS with - i multiplier like ~ 5-7 for n = 128 and n = 8192. Finally, Example ~ 8 shows that there are examples of these DDS with d greater than 2. Before we look at an example using a nontrivial DDS, we can make two minor modifications in this construction that will give similar parameters. Let Si = EA(q ~+ 1) _ H~, the complement of the i 'h hyperplane in EA(q e+ 1). The (trivial) DDS that we have been using for this construction is A = {1} w (H -- K). Ifa 1 = l, then form 600 J.A. DAVIS ARCH. MATH. the set D = H1 u ( 0i=2 aiS~). If the group H is abelian, then we can analyze this set using character theory. If Z is a nonprincipal character on EA (qe+ 1), then Z will be nonprincipal on all the hyperplanes except 1, so the character sum will be 0 on all of those other hyperplanes. Since )~ is nonprincipal on EA(q a+ 1), it will haVe character sum of 0 on all of the S~ associated to those hyperplanes. On the one hyperplane that Z is principal on, the character sum will have modulus the size of the hyperplane over either H~ or ~., which is qd. If Z is principal on EA(q a+ 1), but nonprincipal on K, then I claim that the sum over h (.j a~S~ will be 0. Thus, the sum over D is the size of H 1, which is q! Finally, if Z is i=2 principal on KxEA(q e+l) but nonprincipal on H, then the character sum is qa(n(q - 1) - !). This proves the following theorem. Theorem 3.2. There is a (m, nqd+l, qd(qa+l _ q + i), qd(qa _ i) (q -- 1), q d ~(q -- t) (q(qa _ 1) -- n(q -- 1) + 2))DDS in H x EA(q a+ 1) relative to K x EA(qd+ 1). I am not aware of any new ADS from this theorem, but there will be new reversible DDS similar to the ones in the table above, such as (4, 108, 225, 144, 108); (8, 2744, 16513, 14112, 12180); and (16, 1944, 19521, 12960, 12204). d+ 1 _ 1 One other modification that we can make is to consider any divisor of q : + 1, qd+l 1 q , 1 say n. If ran " +1, and if His any group of order mnwitha normal sub- q-I group K of order n, then label the elements of H as h o, hi,..., hm,-1 where the elements of K are the first n elements in this list. The set D = \i=1 hiH w ~=Jv hi is a (m, nq d+ 1, qa(( m _ 1) n(q - 1) + (n - 1)), qd((m -- 1) n(q -- 1) + (n -- t) -- qd), 21 _ qa-l(q _ 2) [n(q -- 2) + 2]) DDS in H x EA(q ~+ 1) relative to K x EA(q d§ ~). If H is abelian, the easiest proof is character theoretic like the above proof (we d ~ not include it here). There are reversible DDS with these parameters when q = 5, d = 2. Theorem3.3. There arereversibleDDS with the parameters(32,125,3100,2475,2400); (16, 250, 3025, 2400, 2280);(8, 500, 2875, 2250, 2040);(4, 1000, 2575, t950, 1560); and (2,2000,1975,1350,600). The first of these is the complement of one of the DDS found in [2]; I think that the rest are new. Finally, we consider applications of Theorem 3.1 that use nontrivial DDS with k- 21 = 1. In [3], the authors list all known nontrivial DDS with that property, and the list includes two families. The first set has the parameters (q,t n, I + n(q' -- 1_)/2, n(q r __ 1)/2, i + n(q' __ 3)/4) for q' a prime -- 3 (mod4) and n _> 2. The second set has parameters (q', 2, q', q' - 1, (q' - 1)/2) for q' a square of an odd prime qd + 1 _ 1 power. In either of those cases, we want the size of the DDS to divide . We wili state the result, then give examples of the possible constructions, q - 1 Vol. 59, 1992 Almost difference sets 601 ,..-, Corollary 3.5. a. (f q - 1 = I od (for q a prime power and q' a prime power - 3 (rood 4)), then there is a (q',nq'+',qa[qe+ti I-),q' (qa-~), qa- t (I + n@)) DDS in the group EA(q') x K x EA(q d+ 1) relative to K x EA(q d+ 1). qe+l _ 1 b. If is equal to a square of an odd prime power q', then there is a q-I q--1 ' \ q--I '" \~i---i)' DDS in EA(q') x Z z x EA(q d+ l) relative to Z 2 x EA(qa+ l). Examples of the first include a (7, 8, 6, 2, 3) ADS and qd+l_t 3"~"~ 3, qa+2q a-1 qdq 1-1 qaq-1,qa-1 l +qq- q--I q 21' q L i ' -q~ q -- I 4 )) for any prime power q. An example of the second would be (121,486, 980L 3240, t620). References [1] K. T. ARASU,J. A. DAVIS, J. JEDWAB, and S. K. SEHGAL,New Constructions of Menon differ- ence sets. J. Combin. Theory Ser. A, to appear. [2] K. T. ARASU,D. JUNGNICKEL, and A. POTT, Divisible difference sets with m_ultiplier - 1. J. Al- gebra 133, 35-62 (1990). [3] K. 2". ARASU, D. JUNGNICKEL, and A. POTT, Symmetric divisible designs with k - 21 = 1. Discrete Math., to appear. [4] T. BETH, D. JUNGNICKEL, and H. LENZ, Design Theory. Cambridge 1986. [5] J.A. DAVIS, New constructions of divisible designs. Submitted. [6] J. F. DILLON, Variations on a scheme by McFarland for noncyctic difference sets. J. Combin. Theory Set. A 40, 9-2t (1985). [7] D. JUNCNrCKEL, On automorphism groups of divisible designs. Canad. J. Math., 34, 257-297 (1982). [8] D. JUNGNICK~L,A new family of relative difference sets. J. Combin. Theory Ser. A 52, 301-303 (1989). [9] E. S. LANDER, Symmetric designs: an algebraic approach. Cambridge I983. [10] S. L. MA, On divisible difference sets which are fixed by the inverse. Arch. Math. 54, 409-416 (1990). [1!] S. L. MA, It Abelian divisible difference sets with multiplier -1, J. Combin. Theory Set. A, to appear. [12] R. L. MCFARLAND,A family of difference sets in noncyclic groups. J. Combin. Theory Ser. A 15, 1-]0 (1973). 602 J.A. DAVIS ARCH. MATH. [13] R.J. THRYN, Character sums and difference sets. Pacific J. Math. 15, 319-346 (1965). [14] R. J. TURYN, A special class of Williamson matrices and difference sets. J. Combin. Theory Ser. A 36, 111-115 (1984). Eingegangen am 16.8. 1991 *) Anschrift des Autors: James A. Davis Department of Mathematics and Computer Science University of Richmond Richmond, Virginia 23173 USA *) Eine Neufassung ging am 20. 3. 1992 ein.